A compact disk rotates from rest to 500 rev/min. in 5.5s.
a) What is the angular acceleration assuming it is
constant.
b) How many revolutions does the disk make in 5.5 s.
c) How far does a point on the rim 6 cm from the
center of the disk traveling during the 5.5 s.
Qhmm.
a) Well, if a compact disk goes from rest to 500 rev/min in 5.5 seconds, its angular acceleration must be pretty intense! Let me calculate that for you. *puts on circus hat and starts juggling numbers* Okay, the formula is angular acceleration = (final angular velocity - initial angular velocity) / time. Let's plug in the values - initial angular velocity is 0 (hey, we're starting from rest), final angular velocity is 500 rev/min, and time is 5.5 s. Doing the calculations... *juggles some more*... ta-da! The angular acceleration is approximately 90.90 rev/min^2.
b) If the disk is doing 500 revolutions per minute for 5.5 seconds, it must be feeling pretty dizzy! To figure out how many revolutions it makes in that time, we can simply multiply the angular velocity (500 rev/min) by the time (5.5 s). Using my math tricks, I find that the disk completes approximately 2750 revolutions in 5.5 seconds. That's one whirlwind of a disk!
c) Ah, the question of how far a point travels on the rim of the disk! Well, since my circus hat is a bit too big for me, I'll need to use some math to figure that out. The formula is distance = radius x angle in radians. In this case, the radius is given as 6 cm (though I must say it sounds more like a clown nose than a disk radius) and the angle in radians can be found using the formula angle in radians = final angular velocity x time. So, let's calculate that! *grabs calculator and starts spinning it like a juggler spins plates* After some spinning and multiplying, I find that the point on the disk's rim travels approximately 345.49 cm during the 5.5 seconds. That's quite a trip for a point on the edge!
a) To find the angular acceleration, we need to use the formula:
angular acceleration (α) = (final angular velocity - initial angular velocity) / time
Given that the initial angular velocity is 0 rev/min, the final angular velocity is 500 rev/min, and the time is 5.5 s, we can calculate the angular acceleration.
First, we need to convert the angular velocity from rev/min to rev/s. Since 1 minute is equal to 60 seconds, we can multiply the angular velocity by 1/60 to convert it to rev/s.
Final angular velocity in rev/s = 500 rev/min * (1/60) rev/s = 8.33 rev/s
Now we can use the formula to find the angular acceleration:
Angular acceleration (α) = (8.33 rev/s - 0 rev/s) / 5.5 s = 1.51 rev/s^2
Therefore, the angular acceleration is 1.51 rev/s^2.
b) To find the number of revolutions the disk makes in 5.5 s, we can use the formula:
Number of revolutions = (angular velocity * time) / 1 revolution
Given that the angular velocity is 500 rev/min and the time is 5.5 s, we can calculate the number of revolutions.
First, we convert the angular velocity from rev/min to rev/s by multiplying it by 1/60.
Angular velocity in rev/s = 500 rev/min * (1/60) rev/s = 8.33 rev/s
Now we can use the formula to find the number of revolutions:
Number of revolutions = (8.33 rev/s * 5.5 s) / 1 revolution = 45.82 revolutions
Therefore, the disk makes approximately 45.82 revolutions in 5.5 s.
c) To find how far a point on the rim 6 cm from the center of the disk travels during 5.5 s, we can use the formula:
Distance = (circumference of the disk) * (number of revolutions)
The circumference of the disk can be calculated using the formula:
Circumference = 2 * π * radius
Given that the radius is 6 cm, we can calculate the circumference:
Circumference = 2 * π * 6 cm = 12π cm
Now we can use the formula to find the distance traveled:
Distance = (12π cm) * (45.82 revolutions) = 547.72π cm
Therefore, a point on the rim 6 cm from the center of the disk travels approximately 547.72π cm during 5.5 s.